Proof and proving in school mathematics instruction: Making the elementary grades part of the equation
Stylianides, A.J.
PhD Thesis, University of Michigan, Ann Arbor, 2005
The concept of proof and the practice of proving have not typically been a
focus of attention by those involved in the teaching and learning of mathematics
in the elementary grades. This constitutes a serious lack in the way we conceptualize
elementary school mathematics. Because proof and proving are at the core of
doing and knowing mathematics, we cannot have a viable elementary school mathematics
curriculum, or opportunities for students to learn it that have integrity, without
having a way to incorporate proof and proving into a coherent conception of
mathematics instruction in the early grades. This study investigates (1) how
proof and proving might be conceptualized in the context of elementary mathematics
instruction, and (2) how this conceptualization can inform the work that elementary
teachers would need to do, and the knowledge that they would need to have, to
promote proof and proving in their classrooms.
The study is structured around three interrelated strands of work. The first
strand illuminates the nature of proof in the early grades by identifying and
exploring parameters potentially determinant of which arguments qualify as proofs.
The second strand capitalizes on the first and sets forth a framework about
the meaning of proof in K-12 mathematics instruction. Additionally, it uses
this framework as a tool to examine instruction in the early grades in order
to clarify aspects of teachers' role in fostering proof and proving. As a result,
the second strand also develops a framework about instructional practices for
cultivating proof and proving that make sense even in the early grades. The
third strand investigates the knowledge of proof needed for cultivating proof
and proving in elementary mathematics instruction. It advances a classification
of different kinds of knowledge of proof that elementary teachers might need,
paying particular attention to what is involved for teachers as they mobilize
opportunities for their students to engage in proving. The study pursues the
three strands of work both conceptually, using scholarly work on proof (including
work on mathematical practice), and empirically, using data from the teaching
practice of an elementary teacher who was trying to cultivate her students'
reasoning skills.
The products of the study offer insight into what it might mean, and what it
would take, to make proof and proving central to elementary children's learning
of mathematics. The conceptual analytic tools developed by the study contribute
to theory building in the teaching and learning of proof and proving in the
early grades but also more broadly. Furthermore, these tools can support the
design of materials for the professional education of elementary teachers, and
provide guidance for mathematics teacher educators who might implement them